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triprime-comma

The triprime commas of a rank two temperament are the reduced commas of the temperament (meaning they are greater than one and not a power) which factor into three primes or less. The coefficients of the monzos of the triprime commas can be taken from the coefficients of the wedgie for the temperament up to determination of the sign. Hence, they may be computed very quickly and easily from the wedgie. They have various uses; for instance they can be used to find a comma basis for the temperament.

Rank n temperaments always come equipped with n+1 prime factor commas. For instance, 5-limit 12-et has 3^12/2^19, 2^7/5^3 and 3^28/5^19 as two-prime commas, which you can see directly derive from the val, <12 19 28|. A rank three temperament likewise has four prime commas, etc.

The commas in the above example appear in monzo format thus:

   2  3    5

|-19 12,   0 >
|  7  0,  -3 >
|  0 28, -19 >
		

Suppose W is a bival in some prime limit q.

• It has a basis we may write as b_[pi, pj] where pi, pj are primes <= q;
• the coefficient of W with basis b_[pi, pj] we may call w_[pi, pj].
• If p = ithprime(n), we define e(p) as +1 if n is even, and -1 if n is odd; hence e(2) = -1, e(3) = +1, e(5) = -1 and so forth.

Suppose p1 < p2 < p3 are three primes <= q;

• let sign(n) be the signum function, +1 if n is postive, and -1 if n is negative;
• in other words |n|/n, and let e(a,b,c) = sign((a-b)*(b-c)*(c-a)).

Then C = p1^(e(p2,p3,p1)*w_[p2, p3]) * p2^(e(p1,p3,p2)*w_[p1, p3]) * p3^(e(p1,p2,p3)*w_[p1, p2])

If C<1 we take the inverse, and if it is a power we reduce it by taking the gcd of the exponents and dividing it out; in other words if it is an nth power we take the nth root of it.

The result is the {p1, p2, p3} -triprime comma of the wedgie W; taken together these are the triprime commas of W.

It should be noted that sometimes the comma is simply a 1; however any comma of W is a product of its triprime commas.

For trivals you would similarly get quadprime commas, and so forth; and obviously for vals we can easily define the biprime commas.

Example:

From the 7-limit meantone wedgie <<1 4 10 4 13 12||, we get the four triprime commas:

factors      monzo

{3,5,7}    |  0 12 -13 4 >
{2,5,7}    |  6  0 -5  2 >
{2,3,7}    |-13 10  0 -1 >
(2,3,5}    | -4  4 -1  0 >
		

Going up to the 11-limit would add six more such commas; for {2,3,11}, {2,5,11}, {2,7,11}, {3,5,11}, {3,7,11} and {5,7,11}, which of course would be different for the two different versions of 11-limit meantone [meanpop and huygens].

[from Monzo: Note the diagonal of zeros which runs down the matrix of monzos in the example above.]

... we think of 7-limit temperaments sharing a 5-limit comma as belonging to the same extended family, and the name family if they have corresponding periods and generators. This is a stronger relationship than merely sharing a comma, since it means that if you reduce down to the {2,3,5} temperament, they are the same.

However, sharing any tricomma will do something similar.

For instance:

Hemithirds is another temperament you could play this game with, boosting your meantone piece to 118-equal.